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- Title
Asymptotic properties of random matrices of long-range percolation model.
- Authors
Ayadi, S.
- Abstract
We study the spectral properties of matrices of long-range percolation model. These are N × N random real symmetric matrices H = { H( i, j)} i,j whose elements are independent random variables taking zero value with probability , where ψ is an even positive function with ψ( t) ≤ 1 and vanishing at infinity. We study the resolvent G( z) = ( H – z)–1, Im z ≠ 0, in the limit N, b → ∞, b = O( Nα), 1/3 < α < 1, and obtain the explicit expression T( z1, z2) for the leading term of the correlation function of the normalized trace of the resolvent gN,b( z) = N–1Tr G( z). We show that in the scaling limit of local correlations, this term leads to the expression found earlier by other authors for band random matrix ensembles. This shows that the ratio b2/ N is the correct scale for the eigenvalue density correlation function and that the ensemble we study and that of band random matrices belong to the same class of spectral universality.
- Subjects
RANDOM matrices; PERCOLATION; MATRICES (Mathematics); PROBABILITY theory; RANDOM operators
- Publication
Random Operators & Stochastic Equations, 2009, Vol 17, Issue 4, p295
- ISSN
0926-6364
- Publication type
Article
- DOI
10.1515/ROSE.2009.019